Post # 25 Thinking Vertically - Working with Older Students

For intervention with older students, one needs to think of skills introduced but not mastered.  Begin to think about concepts which bridge multiple operations and levels.  Two examples are "Regrouping" and "Place Value."  With older students who have been taught procedures without concepts this is a terrific place to begin. 

Start by using manipulatives to model whole number operations with a large place value mat.  Ask students to "prove by construction" answers to basic problems without regrouping.  With severe students I would recommend the craft sticks because as I have said, the student may need to physically bundle and unbundle quantities.  If the student needs only to reinforce the concept, base ten blocks may be used.  After regrouping is introduced, practiced and mastered, you can move the student to fraction concepts including regrouping from the whole to the part.

Creation of fractions with your fraction circles is a first step.  Students may keep one circle uncut to remind them of how many pieces it takes to make a "whole."  Other fraction pieces may be used to add and subtract like fractions.  If the solution is an improper fraction, the students quickly see that laying the whole circle on top "simplifies" the fraction to a mixed number.

The next step is helping them understand that we may "regroup" from the one's place value to the fraction place value by moving "one" to the fraction place value and representing it as the required circle "cut" into the required size pieces.  The whole can never be in the fraction place value unless it has been "cut" into its required number of pieces, thus creating an improper fraction in the fraction place value. 

I have also used pattern blocks to model this concept and procedure.  The new place value mat using only the 1’s place and Fractions of 1 is a great place value practice sheet.  You might even create a mat with 10’s, 1’s, and “Fractions of 1.”  Using this mat, you can ask students to model regrouping from 10 to 1’s and then from 1’s to Fractions of 1.   This enables you to link to prior learning very efficiently and move on to regrouping with fractions and decimal fractions.  

The student learns that we may get a "sum" which is improper in any place value by the operation of addition.  We then "simplify" the quantity to its proper form.  We may need to create an improper quantity in ANY place value in order to subtract.  This is a fundamental concept for both whole number operations and fractions.

The older student feels validated in that he or she is working at higher levels of math, but is also beginning to understand fundamental math concepts which form the foundations of higher level skills.

I also want to emphasize the need for fluency and familiarity with multiplication facts.  Too many older students were given calculators as an accommodation at which point teachers stopped having them devote time to developing fluency.  For students with language based disabilities, multiplication fact memorization has been demonstrated to be largely a language retrieval problem not a math disability.  These students may take a very long time to develop a degree of fluency which can support reasoning. This is not to say they cannot develop it, simply that it can take a very long time.

By offering targeted practice, especially in what I call “high value products” students can continue to work toward the fluency they need.  By high value products, I mean those which have several factors.  They are often the ones students must factor in order to simplify fractions or expressions at many levels of math. Think twelve, twenty-four, thirty-six, forty-eight, even seventy two. Students will see these numbers again and again throughout fraction studies and algebra. Repeated, targeted practice to automaticity is one ticket to independence.